The same principle applies to the selection of laboratory instrumentation. I've often used 5-1/2 digitbenchtopmultimeters (the "1/2" means that the maximum count is 199999), but it's rare when the 3-1/2 digit multimeter I purchased for about $20 doesn't give the information I need. Digitalpan balances with five or six digits are very convenient, but most of the weighing I've done in my career didn't need such accuracy. Such was true throughout history when simple balance scales were used in commerce and later in science.

The accurate measurement of mass has been important to humans since antiquity, and balance scales with their associated balancing weights have existed for more than 4,000 years. The balance scale was so important to human culture that it appears as the constellation, Libra, the Latinword for the balance scale (see figure).

With such knowledge it's possible to apply a conservation law to the orbit of the Earth to calculate the mass of the Sun. The gravitational force holding the Earth and Sun together must equal the centripetal force that keeps the Earth in its orbit around the Sun. For a circular orbit of radiusr, the centripetal force is

where ME is the mass of the Earth, and v is the orbital velocity, easily calculated from the orbital radius and the length of the year. Once we have the mass of the Sun, it's possible to calculate the mass of the planets through similar calculations. Planetary orbits are elliptical, rather than circular, so the centripetal force would vary, but so would the orbital radius r and the gravitational force. A circular orbit is just a special case of an elliptical orbit that makes such calculations easy.

It's easy to sum the masses of the Sun and its planets, but it's harder to calculate the total mass of the Solar System, since the masses of the Kuiper Belt and Oort Cloud can only be estimated. The vast majority of Solar System mass is in the Sun, since the total mass of Kuiper Belt objects is estimated to be less that a tenth that of the Earth, and the Oort Cloud's mass is about five times Earth's mass.

The next step up on the cosmological scale is the Milky Way Galaxy, which may contain up to 400 billion stars. One way to estimate the mass of the Milky Way would be to assume that our Sun is an average star, and just multiply the mass of the Sun (333,000 Earth masses, or 1.9885 x 1030kilograms) by 400 billion to obtain 2 x 1041 kilograms. That estimate would be wrong, since we've ignored its contained dark matter, and the supermassive black hole at its center. This estimate is a factor of ten too small.

Hubble and Gaia were used to measure the movements in three dimensions of globular star clusters, isolated patches of hundreds of thousands of stars that orbit the Milky Way.[2-3] About fifty of these star clusters were formed before creation of the Milky Way's spiral disk, and they can be used to measure the Milky Way's mass.[2-4] Says Tony Sohn of the Space Telescope Science Institute, who led the Hubble measurements, "Because of their great distances, globular star clusters are some of the best tracers astronomers have to measure the mass of the vast envelope of dark matter surrounding our galaxy far beyond the spiral disk of stars."[2-3] The Hubble-Gaia study was based on a decade of observations of 34 globular clusters out to 65,000 light-years and one cluster out to 130,000 light-years.[2-3]

The analysis of the motions of the globular clusters gave a Milky Way mass of 1.28 (+0.97-0.48) x 1012 (about a trillion) solar masses,[1] a mass that makes our galaxy somewhat of a heavyweight, since the lightest galaxies are about a billion solar masses. It's still lighter than some, since the heaviest galaxies are about 30 trillion solar masses.[2-3] In any case, it's assumed that most of its mass is in the form of dark matter.